No. The Amish and Quakers (the Religious Society of Friends) are distinct Christian traditions with no direct religious or cultural connection, even though both are sometimes called peace churches and share values like simplicity and nonviolence.
The Amish grew out of Swiss-German Anabaptism and are known for adult baptism, congregational discipline, plain dress, and limited technology (including horse-and-buggy transportation in Old Order communities). Quakers arose in 17th-century England around the teaching of George Fox and emphasize direct guidance by the Spirit; many unprogrammed meetings worship largely in silence and do not practice water baptism or communion.
PBS’s American Experience notes that Quakers, along with groups such as the Shakers and Moravians, are sometimes confused with the Amish because of overlapping historical practices (pacifism, plain dress, simplicity), but “none of these four groups have any direct religious or cultural connections with the Amish.”